A tree that is 100 feet tall casts a shadow that is 150 feet long. Determine the angle at which the rays of the sun hit the ground, to the nearest degree.
tan ( theta ) = 100 / 150 = 2 * 50 / ( 3 * 50 )
tan ( theta ) = 2 / 3
theta = tan ^ - 1 ( 2 / 3 )
theta = 33.69 °
theta = 34 ° ( to the nearest degree )
Remark :
tan ^ - 1 ( 2 / 3 )
is the inverse tangent function of 2 / 3
To determine the angle at which the rays of the sun hit the ground, we can use the properties of similar triangles. The height of the tree forms one side of a right triangle, and the length of the shadow forms another side. The angle formed by these two sides is the angle of interest.
Let's call the angle we want to find θ. According to the properties of similar triangles, the ratio of the length of the shadow to the height of the tree is the same as the ratio of the adjacent side to the hypotenuse in a right triangle with angle θ.
In other words, we have the equation:
tan(θ) = length of shadow / height of tree
Substituting the given values, we get:
tan(θ) = 150 / 100
Now, we need to find the value of θ. To do that, we will take the inverse tangent (arctan) of both sides of the equation:
θ = arctan(150 / 100)
Using a calculator, we can find the value of arctan(150 / 100) to be approximately 56.31 degrees.
Therefore, the angle at which the rays of the sun hit the ground, to the nearest degree, is 56 degrees.
To determine the angle at which the rays of the sun hit the ground, we can use trigonometry. The tangent of an angle is equal to the opposite side (the height of the tree) divided by the adjacent side (the length of the tree's shadow).
In this case, the opposite side is the height of the tree, which is 100 feet, and the adjacent side is the length of the shadow, which is 150 feet.
So, we can calculate the angle as:
angle = arctan(opposite/adjacent)
= arctan(100/150)
Using a calculator, we find that:
angle ≈ 33.7 degrees
Therefore, the angle at which the rays of the sun hit the ground is approximately 33.7 degrees.